A probabilistic approach to Euclidean Dirac fields

Abstract

Using a simple isomorphism between the symmetric and antisymmetric Fock space over L2(Rd) and the Wiener isomorphism, a unitary mapping U(s)𝒜ℬ:𝒜(ℋs)→L2R(𝒮′R(Rd), dφsR) is constructed, yielding an isomorphism Ws,l■𝒜(ℋsl)⊗𝒜(ℋsl)→L2R (𝒮C(Rd)l,dlμsR), where ℋs Sobolev space of order s, dφs Gaussian measure on the real Schwartz distributions with covariance operator (−Δ+1)s, dμs=dφs⊗dφs. Using W−1/2,4, the fields are constructed on L2(𝒮′C(R4)4,d4μ−1/2R), yielding a functional integral representation for the two point function of the Euclidean Dirac field with a true Gaussian measure.